The Mackey–Arens theorem is an important theorem in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
that characterizes those
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
vector topologies that have some given space of
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
s as their
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
.
According to Narici (2011), this profound result is central to
duality theory
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the du ...
; a theory that is "the central part of the modern theory of topological vector spaces."
Prerequisites
Let be a vector space and let be a vector subspace of the algebraic dual of that
separates points
''Separates'' is the second album by English punk rock band 999, released in 1978. ''Separates'' was released in the United States under the title ''High Energy Plan'', with a different cover and slightly altered track listing; on ''High Energ ...
on .
If is any other locally convex Hausdorff topological vector space topology on , then we say that is compatible with duality between and if when is equipped with , then it has as its continuous dual space.
If we give the weak topology then is a Hausdorff locally convex
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) and is compatible with duality between and (i.e.
).
We can now ask the question: what are ''all'' of the locally convex Hausdorff TVS topologies that we can place on that are compatible with duality between and ?
The answer to this question is called the Mackey–Arens theorem.
Mackey–Arens theorem
See also
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Dual system
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb.
Duality theory, the study of dual s ...
*
Mackey topology In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not mak ...
*
Polar topology
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a pairing ...
References
Sources
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{{DEFAULTSORT:Mackey-Arens theorem
Theorems in functional analysis
Lemmas
Topological vector spaces
Linear functionals